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2001,Bio4118AppliedBiostatistics,L5.1,Lecture5:Standarderrorandthefundamentalprincipleofthet-test,Whatisastandarderror?Theunderlyingprincipleofthet-testHypothesistestingusingthet-testTestassumptions,2001,Bio4118AppliedBiostatistics,L5.2,Whatisastandarderror?,ApproximatelytheaveragedeviationamongtheresultsiftheexperimentwererepeatedmanytimesunderidenticalconditionsItis,therefore,ameasureofprecision.Itcanbecalculatedforanyparameter/statistic,e.g.themean,thevariance,theslopeofaregression,etc.,2001,Bio4118AppliedBiostatistics,L5.3,Anexample:thestandarderrorofthemean,HistogramdepictingthemeansmiofN=1400randomsamplesof5houseflywinglengthsstandarderrorofmean(SEM)givenby:,2001,Bio4118AppliedBiostatistics,L5.4,Howtocomputeastandarderror,Moredifficult:repeattheexperimentalargenumberoftimesforeachtrial,generateanestimateofthestatistic/parameterofinterestcalculatetheabsolutevalueoftheaveragedeviationfromthemeanofalltrialsLessdifficult:openastatisticsbook,2001,Bio4118AppliedBiostatistics,L5.5,Theunderlyingprincipleofthet-test,Ifthematchbetweenobservedandexpectedispoorerthanwouldbeexpectedonthebasisofmeasurementprecision,thenweshouldrejectthenullhypothesis.,Forklength,Frequency,2001,Bio4118AppliedBiostatistics,L5.6,Whycorrectforprecision?,Largedifferencesbetweenobservedandexpectedmayoccurbecause(1)measurementsareimprecise,or(2)thehypothesisisfalse,or(3)somecombinationofthetwo.Therefore,toconclude(2),wemustfirsteliminate(1)and(3).,Forklength,Frequency,2001,Bio4118AppliedBiostatistics,L5.7,Principleofthet-test,Ifthedifferencebetweentheobservedandexpectedresultsismuchlargerthantheprecisionofthemeasurement,thensomethingiswrong.Ifthedifferencebetweentheobservedresultsandthoseexpectedunderthenullhypothesisismuchlargerthanthestandarderror,thenthenullhypothesisisprobablyincorrect.,2001,Bio4118AppliedBiostatistics,L5.8,Componentsofthet-test,Nullhypothesis(H0)ObservationsTeststatistic(t)Assumptions,2001,Bio4118AppliedBiostatistics,L5.9,TestthatthemeanofasampleisequaltosometheoreticalvaluemTbycalculating:Whatisprobabilityofobtainingatvalueasdeviantasthatobservedgiventhenullhypothesisistrue?,Testinganextrinsichypothesis,2001,Bio4118AppliedBiostatistics,L5.10,Anexample:growthrateinrainbowtrout,Useobservedrelationshipbetweengrowthrate(l)andpHtopredictlinalakeofapH=4.5.NullhypothesisisH0:Compareexpectedl(=lT)withaverageobservedlinlakewithpH=4.5.,2001,Bio4118AppliedBiostatistics,L5.11,Powerandsamplesizeinone-sampletestsofthemean,Supposewewanttodetectadifferenceofatleastdbetweenpredictedmean(m0)andasamplemean(m1).Totestattheasignificancelevelwith1-bpower,wecancalculatetheminimalsamplesizenminrequiredtodetectd,givenanestimatedvariances2.,2001,Bio4118AppliedBiostatistics,L5.12,Powerandsamplesizeinone-sampletestsofthemean,Whatistheminimaldetectabledifferencedminthatcanbedetectedattheasignificancelevelwith1-bpower,givenanestimatedvariances2?,2001,Bio4118AppliedBiostatistics,L5.13,Powerandsamplesizeinone-sampletestsofthemean,SupposeweperformatestandacceptH0.Whatisthepowerofthetest?Givena,d,nands2,cancalculatetb(1)n,andthenusecriticalvaluesofttofindbapproximately,orapproximatetb(1),nbyZb(1).,2001,Bio4118AppliedBiostatistics,L5.14,Knowingn,t,wecanestimaterangeforbandhence1-b.E.g.ifn=2andt=2.1,then.05b.10,so.901-b.95.Alternatively,estimatebbyso1-b=.983,whichisanoverestimateoruseSYSTATpower=tcf(t,n)=.915,Estimatingbfromcriticalvaluesoft,2001,Bio4118AppliedBiostatistics,L5.15,Testinganintrinsichypothesis,Twosamples(1,2)withmeanbodysizesthatdifferbysomeamountd.IfH0:d=0istrue,thentheexpecteddistributionoftheteststatistictis,2001,Bio4118AppliedBiostatistics,L5.16,Anexample(contd),Forthetwopopulationssupposet=2.01.WhatistheprobabilityofgettingavalueatleastthislargeunderH0thatthetwomeansareinfactequal?Sincepissmall,itisunlikelythatH0istrue.Therefore,rejectH0.,2001,Bio4118AppliedBiostatistics,L5.17,Inference:Howtotranslatepintoaconclusion?,Ifp0.05,rejectthenullhypothesis.butkeeppinmind!Reportp,notjustwhetheritis“significant”(ornot).Remember,thep0.05isentirelyarbitrary!,2001,Bio4118AppliedBiostatistics,L5.18,Assumptions,piscalculatedassumingtheteststatistictisdistributedasStudentst(ts)whichhasawell-knowndistribution.Thisassumptionistrueonlyifthedataarenormallydistributed.,2001,Bio4118AppliedBiostatistics,L5.19,ThedistributionoftversusStudentst(ts),Calculationofpassumesp(t)=p(ts).But,asdatabecomeincreasinglynon-normal,thedeviationbetweenthetwoincreases.Therefore,calculatedpvaluesareincorrect.,t,datahighlynon-normal,t,dataslightlynon-normal,ts,2001,Bio4118AppliedBiostatistics,L5.20,Whatifdataarenotnormal?,Translationoftintopisincorrect.But,biasisoftenverysmall,especiallywithlargesamples,duetoCentralLimitTheorem.So,usecommonsense.andworryonlywhenpisclosetonominalalevel.,2001,Bio4118AppliedBiostatistics,L5.21,Increasesamplesize.Transformdata.Useanother(non-parametric)test,onethatdoesnotassumenormality.,Whatifdataarenotnormalandpisclosetoa?,2001,Bio4118AppliedBiostatistics,L5.22,Datatransformations,typicallysimpl
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